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Then interpret the usual propositions about <math>p, q\!</math> as functions of the concrete type <math>f : P \times Q \to \mathbb{B}.</math>
Then interpret the usual propositions about <math>p, q\!</math> as functions of the concrete type <math>f : P \times Q \to \mathbb{B}.</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math>
We are going to consider various "operators" on these functions.
Here, an operator W is a function that takes one function f into
another function Wf.
The first couple of operators that we need to consider are
The first couple of operators that we need to consider are logical analogues of the pair that play a founding role in the classical finite difference calculus, namely:
logical analogues of the pair that play a founding role
in the classical "finite difference calculus", namely:
{| align="center" cellpadding="6" width="90%"
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
|-
| The ''enlargement" operator'' <math>\Epsilon,\!</math> written here as <math>\operatorname{E}.</math>
|}
These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\operatorname{E}X</math> is constructed according to the following specifications:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{rcc}
\operatorname{E}X & = & X \times \operatorname{d}X
\end{array}</math>
|}
where:
{| align="center" cellpadding="6" width="90%"
|
<math>\begin{array}{rcc}
X
& = &
P \times Q
\\[4pt]
\operatorname{d}X
& = &
\operatorname{d}P \times \operatorname{d}Q
\\[4pt]
\operatorname{d}P
& = &
\{ \texttt{(} \operatorname{d}p \texttt{)},~ \operatorname{d}p \}
\\[4pt]
\operatorname{d}Q
& = &
\{ \texttt{(} \operatorname{d}q \texttt{)},~ \operatorname{d}q \}
\end{array}</math>
|}
<pre>
The interpretations of these new symbols can be diverse,
The interpretations of these new symbols can be diverse,
but the easiest interpretation for now is just to say
but the easiest interpretation for now is just to say
